A Sparse ADMM-Based Solver for Linear MPC Subject to Terminal Quadratic Constraint

Model predictive control (MPC) typically includes a terminal constraint to guarantee stability of the closed-loop system under nominal conditions. In linear MPC, this constraint is generally taken on a polyhedral set, leading to a quadratic optimization problem. However, the use of an ellipsoidal terminal constraint may be desirable, leading to an optimization problem with a quadratic constraint. In this case, the optimization problem can be solved using second-order cone (SOC) programming solvers, since the quadratic constraint can be posed as a SOC constraint, at the expense of adding additional slack variables and possibly compromising the simple structure of the solver ingredients. In this brief, we present a sparse solver for linear MPC subject to a terminal ellipsoidal constraint based on the alternating direction method of multipliers (ADMM) algorithm in which we directly deal with the quadratic constraints without having to resort to the use of a SOC constraint nor the inclusion of additional decision variables. The solver is suitable for its use in embedded systems, since it is sparse, has a small memory footprint, and requires no external libraries. We compare its performance against other approaches from the literature.